Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.R.14

Chapter 1, Problem 1.R.14

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>
f⁻¹( g⁻¹(4))

Verified step by step guidance

Understand that an odd function f satisfies the property f(-x) = -f(x) for all x in its domain.

Recognize that if f is one-to-one, it has an inverse function f⁻¹, and similarly for g.

To find f⁻¹(g⁻¹(4)), start by finding g⁻¹(4). This means finding the value x such that g(x) = 4.

Once you have the value of x from g⁻¹(4), use it to find f⁻¹(x). This means finding the value y such that f(y) = x.

Use the properties of odd functions and the given graphs to determine the specific values needed for f⁻¹ and g⁻¹.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Odd Functions

An odd function is defined by the property f(-x) = -f(x) for all x in its domain. This symmetry about the origin implies that the graph of an odd function will reflect across both axes. Understanding this property is crucial when analyzing the behavior of the function and its inverse, especially in relation to the values being computed.

One-to-One Functions

A one-to-one function, or injective function, is one where each output is produced by exactly one input. This means that if f(a) = f(b), then a must equal b. This property is essential for determining the existence of an inverse function, as only one-to-one functions can have well-defined inverses that map back uniquely to their original inputs.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If f is a function and f⁻¹ is its inverse, then f(f⁻¹(x)) = x for all x in the range of f. In this context, finding f⁻¹(g⁻¹(4)) involves first determining g⁻¹(4) and then applying f⁻¹ to that result, highlighting the importance of understanding how to compute and interpret inverse functions.

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Textbook Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure) <IMAGE>.

a. Find the domain and a formula for each function.

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Textbook Question

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = 7 / (x + 3)

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Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.

sin⁻¹ ( -1 )

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Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f^-1(1 + f(-3))

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Textbook Question

Solving equations Solve each equation.

ln 3x + ln (x + 2) = 0

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Textbook Question

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = x² - 2x

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